I do not understand what they are asking or how to set the problem up, so I can

Question

I do not understand what they are asking or how to set the problem up, so I can eventually sove it. Can anyone help?

Problem:

Let Z be the integer. Define the parity relationship on Z x Z by: p is related to q, if and only if, 2|(p-q) (i.e., 2 divides p-q) show that parity is an equivalence relation. What are its equivalence classes?

I understand Z x Z is Z(sub 1) = {1,2,3,4,5,...} and Z(sub 2) = {1,2,3,4,5,...} and since Z(sub1) is equal to Z(sub2) that their relationship is reflective but that is all I understand.

Explanation / Answer

They are reflexive: Z is related to Z, element is related to itself. In other words, a relation ~ on a set S is reflexive when x ~ x holds true for every x in S, formally: when xS: x~x holds.

A refleive relation is a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B.

Examples of reflexive relations include:

"is equal to" (equality)
"is a subset of" (set inclusion)
"divides" (divisibility)
"is greater than or equal to"
"is less than or equal to"

For example as you understand; Let's take a portion of the relation R on {1,2,3,4,5,...} given by {1,2,3} so R = {(1,1), (2,2), (2,3), (3,3)} that's reflexive. (All loops are present)

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